A woman is running towards us, mouth open in a scream, a baby cradled in her arms. The violence around her seems to part and give safe passage through the slaughter. What the open pathway through the heart of the horror really gives, however, is a heartbreaking visual connection between our eyes and her pain. To look into that terrified face is to feel the full pity of her plight. It is impossible not to be gripped by compassion.

There are three drawings of this harrowing New Testament scene, The Massacre of the Innocents, in the Ashmolean Museum’s outstanding exhibition of Raphael’s drawings. They were done in Rome in about 1509-10, when the artist was in his mid-20s. In each – from rough sketch to finished design – the same woman rushes through the crowd. Yet the details vary: the expression on her face, the pose of the baby. In the most poignant, the baby’s eyes are little dots and it lolls as if dead in her arms. Her eyes are hollow dark pits of despair.

I found myself thinking of the dead baby in its screaming mother’s arms in Picasso’s Guernica. This is no coincidence. Picasso profoundly admired Raphael and the way he painted the pity of war in Guernica owes a lot to his study of Raphael. Yet Raphael’s drawings of the Massacre of the Innocents never resulted in a finished painting. Instead, the design was engraved as a print by his collaborator Marcantonio Raimondi. That too is here, and it reveals why so many people today find it hard to share the excitement this Renaissance genius inspired in Picasso.

Raphael’s drawings of The Massacre of the Innocents work through a balance of passion and restraint that lets him depict cruelty without glorifying it, and make the women and children the heroes of this scene. Yet in the print, the discipline of his design seems much colder: instead of breaking your heart, it impresses with its formal rigour.

The print was published in about 1511 and for the next four centuries, Raphael would be studied everywhere as the model of classic perfection. Every young artist had to imitate his mathematically pure compositions. The reason Picasso knew his Raphael so deeply is that he got that kind of old-fashioned art school training in late 19th-century Spain. Today, Raphael seems to many eyes not so much a paragon as boringly strait-laced.

But that image is blown away by this exhibition, which cuts past all the copies, fame and cliche to make contact with the man himself. We stand at his shoulder while he works. We see him come up with ideas. Raphael’s drawings are the direct evidence of his hand and eye, how he saw life, how he put his feelings about it on paper. The man we meet is not a careerist favourite of the pope. He is a rare spirit whose art is a mirror of his idealism.

Raphael’s tenderness glows in his exact yet sensitive line. His drawings of mothers with babies are among his most expressive sketches. Raphael was born in 1483 and orphaned as a child: art was his salvation, giving him a career when he was still a teenager. Did the raw pain of losing his parents young give his drawings of mothers their intensity?

In big red chalk designs for his most powerful fresco The Fire in the Borgo, he imagines how people might behave in a desperate crisis that is consuming a city. As fire sweeps through houses near the Vatican, the people of Rome help each other with loyalty and collective courage. A young man carries his father on his back to get him to safety. Two women try to protect a young child from the heat and falling masonry.

This outstanding show makes you understand why his contemporaries adored Raphael. It was not just that he was very good-looking – as can be seen from a self-portrait at the start of the show – and a famous lover. There’s an innocent sweetness to these drawings, even a goodness.

Not only is he human, he’s vulnerable. We see Raphael here not as some shining cultural monument but a young artist learning on the job. In 1504 he is in Florence, watching a competition between his elders Michelangelo and Leonard da Vinci, copying them both. He happily learns from these two titans. Then he goes to Rome, to paint for popes and cardinals, to rival Michelangelo himself.

Yet his drawing continues to develop. Raphael liked to draw clearly and sharply. Only gradually did he start to experiment with a more sensual, uncontrolled style of sketching in chalk. The most disarming drawing of all was done when he was decorating the villa of the banker Agostino Chigi in about 1517 or 18. According to 16th-century biographer Giorgio Vasari, Raphael was by this time so louche he couldn’t be dragged away from his lover to work on Chigi’s frescoes. In desperation, the banker installed Raphael’s mistress in his house so the artist could go from her bed to his work and back.

Here we see the passionate reality behind that story. Raphael draws not one but three young women posing as the Three Graces. There is something almost embarrassingly intimate about this red chalk sketch. You know it was done from life and you want to know more about the women. Once again, Raphael’s drawing is tender, honest, loving.

Then, suddenly, he was gone. On 6 April 1520 Raphael died, aged 37. The gossip was that he went into a fever after exhausting himself with too much sex. He was mourned like a saint and this exhibition makes you feel the tragedy and loss. His last works are tremendous, his feel for life infectious. This mind-opening show will transform how Raphael is seen.